Online Learning Quantum States with the Logarithmic Loss via VB-FTRL

TL;DR

The paper addresses online learning of quantum states with the logarithmic loss (LL-OLQS), a challenging non-Lipschitz setting that generalizes online portfolio selection to the quantum domain. It extends the VB-FTRL algorithm to LL-OLQS by introducing VB-convexity and a volumetric barrier framework, enabling a convex optimization at each round and a regret analysis analogous to the classical case. The main result is a regret bound of $O(d^2 \log(d+T))$, with each iteration solving a semidefinite program (SDP) via cutting-plane methods, and a detailed regret decomposition showing how the bias, gain, and miss terms contribute. This work advances practical online quantum state estimation and stochastic optimization for quantum tomography by providing a scalable, regret-optimal method with a principled convexity framework and explicit algorithmic steps.

Abstract

Online learning of quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection (OPS), a classic open problem in online learning for over three decades. This problem also emerges in designing stochastic optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first regret-optimal algorithm for OPS with moderate computational complexity. In this paper, we generalize VB-FTRL for LL-OLQS. Let $d$ denote the dimension and $T$ the number of rounds. The generalized algorithm achieves a regret rate of $O ( d^2 \log ( d + T ) )$ for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, for example, cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently $O ( d^2 \log T )$, achieved by an exponential weight method. However, no explicit implementation is available for the exponential weight method for LL-OLQS. To facilitate the generalization, we introduce the notion of VB-convexity. VB-convexity is a sufficient condition for the volumetric barrier associated with any function to be convex and is of independent interest.

Theorems & Definitions (18)

• Theorem 1.1
• Lemma 2.1: Vaidya1996
• Definition 2.1: VB-Convexity
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Corollary 2.6
• Lemma 2.7